Ponder This

July 1998

Wimbledon has just ended, and your job is to store the remaining tennis balls in buckets. The storage area will only hold rows of three buckets, but will hold an undetermined number of rows. You must follow three inviolable rules in storing the tennis balls:

Each bucket must have a different positive number of balls.

Each row must have the same total number of balls in its three buckets.

The total number of balls in each row is as small as possible.

For example, if you were given only two rows of buckets, the setup:

achieves a row total of 11. This is the best possible arrangement for two rows. Problem: Can you give a construction that gives an optimal arrangement that works for an arbitrary number k of rows?

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